Multi-Message Private Information Retrieval: Capacity Results and Near-Optimal Schemes

We consider the problem of multi-message private information retrieval (MPIR) from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> non-communicating replicated databases. In MPIR, the user is interested in retrieving <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> messages out of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR <inline-formula> <tex-math notation="LaTeX">$C_{s}^{P}$ </tex-math></inline-formula> is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol, where the symbols are defined over the same field. For the case <inline-formula> <tex-math notation="LaTeX">$P \geq M/2$ </tex-math></inline-formula>, we determine the exact sum capacity of MPIR as <inline-formula> <tex-math notation="LaTeX">$C_{s}^{P}=1/(1+(M-P)/(PN))$ </tex-math></inline-formula>. The achievable scheme in this case is based on downloading MDS-coded mixtures of all messages. For <inline-formula> <tex-math notation="LaTeX">$P \leq {M}/{2}$ </tex-math></inline-formula>, we develop lower and upper bounds for all <inline-formula> <tex-math notation="LaTeX">$M,P,N$ </tex-math></inline-formula>. These bounds match if the total number of messages <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> is an integer multiple of the number of desired messages <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula>, i.e., <inline-formula> <tex-math notation="LaTeX">$M/P \in \mathbb {N}$ </tex-math></inline-formula>. In this case, <inline-formula> <tex-math notation="LaTeX">$C_{s}^{P}=(1+1/N+\cdots +1/N^{M/P-1})^{-1}$ </tex-math></inline-formula>, i.e., <inline-formula> <tex-math notation="LaTeX">$C_{s}^{P}=(1-1/N)/(1-1/N^{M/P})$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$N>1$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$C_{s}^{P}=P/M$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$N=1$ </tex-math></inline-formula>. The achievable scheme in this case generalizes the single-message capacity achieving scheme to have unbalanced number of stages per round of download. For all the remaining cases, the difference between the lower and upper bound is at most 0.0082, which occurs for <inline-formula> <tex-math notation="LaTeX">$M=5$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$P=2$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$N=2$ </tex-math></inline-formula>. Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes even after considering the free savings that result from downloading undesired symbols in each single-message retrieval round.

[1]  Salim El Rouayheb,et al.  Private Information Retrieval From MDS Coded Data in Distributed Storage Systems , 2016, IEEE Transactions on Information Theory.

[2]  Ian Goldberg,et al.  One (Block) Size Fits All: PIR and SPIR with Variable-Length Records via Multi-Block Queries , 2013, NDSS.

[3]  A. W. M. van den Enden,et al.  Discrete Time Signal Processing , 1989 .

[4]  Hua Sun,et al.  Multiround Private Information Retrieval: Capacity and Storage Overhead , 2016, IEEE Transactions on Information Theory.

[5]  Kannan Ramchandran,et al.  One extra bit of download ensures perfectly private information retrieval , 2014, 2014 IEEE International Symposium on Information Theory.

[6]  Hirosuke Yamamoto,et al.  Private information retrieval for coded storage , 2014, 2015 IEEE International Symposium on Information Theory (ISIT).

[7]  Mikael Skoglund,et al.  Symmetric private information retrieval for MDS coded distributed storage , 2016, 2017 IEEE International Conference on Communications (ICC).

[8]  Sennur Ulukus,et al.  The Capacity of Private Information Retrieval From Coded Databases , 2016, IEEE Transactions on Information Theory.

[9]  William Gasarch.  A Survey on Private Information Retrieval , 2004 .

[10]  Hua Sun,et al.  The Capacity of Robust Private Information Retrieval With Colluding Databases , 2016, IEEE Transactions on Information Theory.

[11]  Amir Herzberg,et al.  RAID-PIR: Practical Multi-Server PIR , 2014, CCSW.

[12]  Yong Liu,et al.  A Fast Multi-Server, Multi-Block Private Information Retrieval Protocol , 2014, 2015 IEEE Global Communications Conference (GLOBECOM).

[13]  Rafail Ostrovsky,et al.  A Survey of Single-Database Private Information Retrieval: Techniques and Applications , 2007, Public Key Cryptography.

[14]  Sennur Ulukus,et al.  The Capacity of Private Information Retrieval from Byzantine and Colluding Databases , 2017, IEEE Transactions on Information Theory.

[15]  Rafail Ostrovsky,et al.  Batch codes and their applications , 2004, STOC '04.

[16]  Eyal Kushilevitz,et al.  Private information retrieval , 1998, JACM.

[17]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[18]  F. Moore,et al.  Polynomial Codes Over Certain Finite Fields , 2017 .

[19]  Ian Goldberg,et al.  Sublinear Scaling for Multi-Client Private Information Retrieval , 2015, Financial Cryptography.

[20]  Hua Sun,et al.  Blind interference alignment for private information retrieval , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[21]  Hua Sun,et al.  The Capacity of Private Information Retrieval , 2017, IEEE Transactions on Information Theory.

[22]  Aggelos Kiayias,et al.  Multi-query Computationally-Private Information Retrieval with Constant Communication Rate , 2010, Public Key Cryptography.

[23]  Camilla Hollanti,et al.  Private Information Retrieval from Coded Databases with Colluding Servers , 2016, SIAM J. Appl. Algebra Geom..

[24]  Syed Ali Jafar,et al.  Blind Interference Alignment , 2012, IEEE Journal of Selected Topics in Signal Processing.

[25]  Hua Sun,et al.  The Capacity of Symmetric Private Information Retrieval , 2016, 2016 IEEE Globecom Workshops (GC Wkshps).

[26]  Hua Sun,et al.  Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length , 2016, IEEE Transactions on Information Forensics and Security.

[27]  Stephen B. Wicker,et al.  Reed-Solomon Codes and Their Applications , 1999 .

[28]  Silvio Micali,et al.  Computationally Private Information Retrieval with Polylogarithmic Communication , 1999, EUROCRYPT.